One-dimensional model and solutions for creeping gas flows in the approximation of uniform pressure

A model, along with analytical and numerical solutions, is presented to describe a wide variety of one-dimensional slow flows of compressible heat-conductive fluids. The model is based on the approximation of uniform pressure valid for the flows, in which the sound propagation time is much shorter than the duration of any meaningful density variation in the system. The energy balance is described by the heat equation that is solved independently. This approach enables the explicit solution for the fluid velocity to be obtained. Interfacial and volumetric heat and mass sources as well as boundary motion are considered as possible sources of density variation in the fluid. A set of particular tasks is analyzed for different motion sources in planar, axial, and central symmetries in the quasistationary limit of heat conduction (i.e., for large Fourier number). The analytical solutions are in excellent agreement with corresponding numerical solutions of the whole system of the Navier-Stokes equations. This work deals with the ideal gas. The approach is also valid for other equations of state.

Figure 1

Gas flow sources in different geometries and typical flow velocity profiles: (a) a cavity with temperature variation on the walls; (b) moving boundary; (c) gas inflow and outflow on the boundaries; and (d) distributed mass source of heat and/or mass.

Figure 2

Flow between two heating plates: (a) Dimensionless velocity profiles for boundary conditions (49) with $\mathrm{\Theta }=\mathrm{\Delta }T/T_0=0.2$: the exact analytic solution (50) (thin solid lines), the UIT approximation (dashed-dotted lines), and the first-order approximation (28) (thick dashed lines) at several time instants (numbers near the curves show the $\omega t$ value). The diamonds locate the velocity extrema. (b) Comparison of the analytical solution (52) (solid line) and three numerical solutions (red diamonds $\mathrm{Fo}=10$, green squares $\mathrm{Fo}=1$, and blue circles $\mathrm{Fo}=0.1$) for boundary conditions (51) at the time instant $\tau =kt/T_0=\frac{1}{15}$.

Figure 3

Flow between two heating cylinders: (a) Profiles of dimensionless velocity at several values of the ratio $G=r_2/r_1$ (numbers near the curves). The profiles are calculated at $t=0$ for the periodic boundary conditions (49) with $\mathrm{\Theta }=\mathrm{\Delta }T/T_0=0.2$. (b) Comparison of the analytical (solid line) and three numerical solutions (red diamonds $\mathrm{Fo}=10$, green squares $\mathrm{Fo}=1$, and blue circles $\mathrm{Fo}=0.1$) in the case of linear boundary conditions (51) at $\tau =kt/T_0=\frac{1}{15}$ and $G=10$.

Figure 4

Boundary temperature variation: (a) Velocity extremum location as a function of decimal logarithm of the ratio $G=r_2/r_1$ in all three geometries for boundary conditions (49) at $\tau =0$ and fixed $r_1$. (b) Comparison of the velocity distributions in different symmetries for linear boundary conditions (51) at dimensionless time $\tau =kt/T_0=\frac{1}{15}$ and $G=10$. Dashed lines are for the UIT approximation.

Figure 5

Heating and moving wall: (a) Analytic profiles (thin solid curves) of dimensionless gas velocity for boundary conditions (73) at $\mathrm{\Theta }=\mathrm{\Delta }T/T_0=\frac{1}{15}$, $\delta =\mathrm{\Delta }r/(r_{20}-r_{10})=0.01$. Thick dashed lines represent the solution for constant temperature of the left wall $T_1=T_0$. The numbers next to the curves show corresponding values of $\omega t$. (b) Dimensionless velocity profiles from Eq. (72) (solid curves) and numerical simulation (symbols), at $t=0$, $\mathrm{\Theta }=\frac{1}{15}$, and three values of $\delta$ (numbers next to the curves). The profiles are normalized by corresponding wall velocity.

Figure 6

Boundary mass sources: (a) Velocity profiles at constant gas density, i.e., for boundary fluxes $j_2={\left(r_1/r_2\right)}^n{j}_1$. (b) Profiles for equal inflow and outflow fluxes $j_1=-j_2\equiv j_0$. The radii ratio is $G=5$ everywhere except for the dashed curve in the plot (b).

Figure 7

Distributed mass sources with Arrhenius-type kinetics: (a) Gas velocity profiles obtained from the exact solution (solid black curve), first approximation (dashed green curve), and numerical simulation in ansys fluent (red circles) at $\alpha =0.3$, $\mathrm{\Theta }=34$ versus dimensionless coordinate $x$ (42). (b) Velocity extremum location (solid lines) and inflection point location (dashed lines) as functions of $\mathrm{\Theta }$ at three values of $\alpha$ denoted next to the curves.

Figure 8

Gas velocity profiles from Eq. (90) for a steplike function of distributed mass source: (a) parametric dependence on the filling factor $r_c/r_2$, the values are next to the curves, plane geometry ($n=0$); (b) parametric dependence on the dimensionality for fixed $r_c/r_2=0.01$. All the velocities are normalized by $\frac{r_{c}RT{J}_0}{(n+1)p}\left[1-{\left(\frac{r_c}{r_2}\right)}^{n+1}\right]$.

Figure 9

Gas flow generated by two small heating spheres inside a spherical cavity at $t=0.05T_0/k$. Hollow circles for numerical velocity profiles and solid lines for superposition of two analytical solutions: (a) radial components $u(r,0)$ along the axis in the equatorial plane and (b) axial components $u(0,z)$ along the axis through the sphere centers.